Theorem summodclem2 11323

Description: Lemma for summodc 11324. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

Hypotheses

Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

Assertion

Ref	Expression
summodclem2	⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Distinct variable groups:   𝑘,𝑛,𝐴   𝑛,𝐹   𝜑,𝑘,𝑛   𝐴,𝑓,𝑗,𝑚,𝑘,𝑛   𝐵,𝑛   𝑓,𝐹,𝑘,𝑚   𝜑,𝑓,𝑚   𝑥,𝑓,𝑘,𝑚,𝑛   𝑦,𝑓,𝑚

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑗)   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚)   𝐹(𝑥,𝑦,𝑗)   𝐺(𝑥,𝑦,𝑓,𝑗,𝑘,𝑚,𝑛)

Proof of Theorem summodclem2

Dummy variables 𝑎 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5486	. . . . 5 ⊢ (𝑚 = 𝑎 → (ℤ≥‘𝑚) = (ℤ≥‘𝑎))
2	1	sseq2d 3172	. . . 4 ⊢ (𝑚 = 𝑎 → (𝐴 ⊆ (ℤ≥‘𝑚) ↔ 𝐴 ⊆ (ℤ≥‘𝑎)))
3	1	raleqdv 2667	. . . 4 ⊢ (𝑚 = 𝑎 → (∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10383	. . . . 5 ⊢ (𝑚 = 𝑎 → seq𝑚( + , 𝐹) = seq𝑎( + , 𝐹))
5	4	breq1d 3992	. . . 4 ⊢ (𝑚 = 𝑎 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑎( + , 𝐹) ⇝ 𝑥))
6	2, 3, 5	3anbi123d 1302	. . 3 ⊢ (𝑚 = 𝑎 → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)))
7	6	cbvrexv 2693	. 2 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥))
8		simplr3 1031	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ 𝑥)
9		simplr1 1029	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ (ℤ≥‘𝑎))
10		uzssz 9485	. . . . . . . . . . . 12 ⊢ (ℤ≥‘𝑎) ⊆ ℤ
11	9, 10	sstrdi 3154	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ⊆ ℤ)
12		1zzd 9218	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 1 ∈ ℤ)
13		simprl 521	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
14	13	nnzd 9312	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
15	12, 14	fzfigd 10366	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
16		simprr 522	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
17		f1oeng 6723	. . . . . . . . . . . . . 14 ⊢ (((1...𝑚) ∈ Fin ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (1...𝑚) ≈ 𝐴)
18	15, 16, 17	syl2anc 409	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (1...𝑚) ≈ 𝐴)
19	18	ensymd 6749	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ≈ (1...𝑚))
20		enfii 6840	. . . . . . . . . . . 12 ⊢ (((1...𝑚) ∈ Fin ∧ 𝐴 ≈ (1...𝑚)) → 𝐴 ∈ Fin)
21	15, 19, 20	syl2anc 409	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝐴 ∈ Fin)
22		zfz1iso 10754	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℤ ∧ 𝐴 ∈ Fin) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
23	11, 21, 22	syl2anc 409	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → ∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
24		isummo.1	. . . . . . . . . . . . 13 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
25		simplll 523	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝜑)
26		isummo.2	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
27	25, 26	sylan 281	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
28		eleq1w 2227	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
29	28	dcbid 828	. . . . . . . . . . . . . 14 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
30		simpr2 994	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
31	30	ad2antrr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴)
32		simpr 109	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → 𝑘 ∈ (ℤ≥‘𝑎))
33	29, 31, 32	rspcdva 2835	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) ∧ 𝑘 ∈ (ℤ≥‘𝑎)) → DECID 𝑘 ∈ 𝐴)
34		summodclem2.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
35		eqid 2165	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑔‘𝑛) / 𝑘⦌𝐵, 0))
36		simprll 527	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑚 ∈ ℕ)
37		simpllr 524	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑎 ∈ ℤ)
38		simplr1 1029	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝐴 ⊆ (ℤ≥‘𝑎))
39		simprlr 528	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
40		simprr 522	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))
41	24, 27, 33, 34, 35, 36, 37, 38, 39, 40	summodclem2a 11322	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ ((𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴))) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
42	41	expr 373	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
43	42	exlimdv 1807	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → (∃𝑔 𝑔 Isom < , < ((1...(♯‘𝐴)), 𝐴) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)))
44	23, 43	mpd 13	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚))
45		climuni 11234	. . . . . . . . 9 ⊢ ((seq𝑎( + , 𝐹) ⇝ 𝑥 ∧ seq𝑎( + , 𝐹) ⇝ (seq1( + , 𝐺)‘𝑚)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
46	8, 44, 45	syl2anc 409	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ (𝑚 ∈ ℕ ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴)) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
47	46	anassrs 398	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑥 = (seq1( + , 𝐺)‘𝑚))
48		eqeq2 2175	. . . . . . 7 ⊢ (𝑦 = (seq1( + , 𝐺)‘𝑚) → (𝑥 = 𝑦 ↔ 𝑥 = (seq1( + , 𝐺)‘𝑚)))
49	47, 48	syl5ibrcom 156	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑦 = (seq1( + , 𝐺)‘𝑚) → 𝑥 = 𝑦))
50	49	expimpd 361	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
51	50	exlimdv 1807	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
52	51	rexlimdva 2583	. . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
53	52	r19.29an 2608	. 2 ⊢ ((𝜑 ∧ ∃𝑎 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑎) ∧ ∀𝑗 ∈ (ℤ≥‘𝑎)DECID 𝑗 ∈ 𝐴 ∧ seq𝑎( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
54	7, 53	sylan2b 285	1 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 103  DECID wdc 824   ∧ w3a 968   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  ⦋csb 3045   ⊆ wss 3116  ifcif 3520   class class class wbr 3982   ↦ cmpt 4043  –1-1-onto→wf1o 5187  ‘cfv 5188   Isom wiso 5189  (class class class)co 5842   ≈ cen 6704  Fincfn 6706  ℂcc 7751  0cc0 7753  1c1 7754   + caddc 7756   < clt 7933   ≤ cle 7934  ℕcn 8857  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  seqcseq 10380  ♯chash 10688   ⇝ cli 11219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220

This theorem is referenced by:  summodc  11324